Hotelling's lemma

Hotelling's lemma is a result in microeconomics that relates the supply of a good to the maximum profit of the producer. It was first shown by Harold Hotelling, and is widely used in the theory of the firm.

Specifically, it states: The rate of an increase in maximized profits w.r.t. a price increase is equal to the net supply of the good. In other words, if the firm makes its choices to maximize profits, then the choices can be recovered from the knowledge of the maximum profit function.

Formal Statement

Let ${\displaystyle p}$ denote a variable price, and ${\displaystyle w}$ be a constant cost of each input. Let ${\displaystyle x:{\mathbb {R} ^{+}}\rightarrow X}$ be a mapping from the price to a set of feasible input choices ${\displaystyle X\subset {\mathbb {R} ^{+}}}$. Let ${\displaystyle f:{\mathbb {R} ^{+}}\rightarrow {\mathbb {R} ^{+}}}$ be the production function, and ${\displaystyle y(p)\triangleq f(x(p))}$ be the net supply.

The maximum profit can be written by

${\displaystyle \pi (p)=\max _{x}p\cdot y(p)-w\cdot x(p).}$

Then the lemma states that if the profit ${\displaystyle \pi }$ is differentiable at ${\displaystyle p}$, the maximizing net supply is given by

${\displaystyle y^{*}(p)={\frac {d\pi (p)}{dp}}.}$

Proof for Hotelling's lemma

The lemma is a corollary of the envelope theorem.

Specifically, the maximum profit can be rewritten as ${\displaystyle \pi (p,x^{*})=p\cdot f(x^{*}(p))-w\cdot x^{*}(p)}$ where ${\displaystyle x^{*}}$ is the maximizing input corresponding to ${\displaystyle y^{*}}$. Due to the optimality, the first order condition gives

${\displaystyle {\frac {\partial \pi }{\partial x}}{\bigg |}_{x=x^{*}}=0.}$

(1)

By taking the derivative by ${\displaystyle p}$ at ${\displaystyle x^{*}}$,

${\displaystyle {\frac {d\pi }{dp}}={\frac {\partial \pi }{\partial x}}{\bigg |}_{x=x^{*}}{\frac {\partial x}{\partial p}}+{\frac {\partial \pi }{\partial p}}={\frac {\partial \pi }{\partial p}}=f(x^{*}(p))=y^{*}(p)}$

where the second equality is due to (1). QED

Application of Hotelling's lemma

Let the firm's profit function be:

${\displaystyle \pi =py-wx}$

where:

• ${\displaystyle \pi }$ is profit
• ${\displaystyle p}$ is the price of output ${\displaystyle y}$
• ${\displaystyle y}$ is output
• ${\displaystyle w}$ is input price for input ${\displaystyle x}$
• ${\displaystyle x}$ is the single input needed for producing ${\displaystyle y}$

If a firm produces 10 units of ${\displaystyle y}$ using 5 units of input ${\displaystyle x}$ which cost 1 dollar each and sells each output for 2 dollars. the profit the firm makes is:

${\displaystyle \pi _{1}=(2)(10)-(1)(5)=15}$

If the firm increases the price of the output to 3 dollars and still sells the same amount of ${\displaystyle y}$, the firm's profits are now:

${\displaystyle \pi _{2}=(3)(10)-(1)(5)=25}$

Taking the difference between ${\displaystyle \pi _{1}}$ and ${\displaystyle \pi _{2}}$

${\displaystyle \Delta \pi =\pi _{2}-\pi _{1}=25-15=10}$

The change in profits from a change in price is 10, which is exactly the same as the output produced. thus the statement of ${\displaystyle y(p)={\frac {\partial \pi (p)}{\partial p}}}$ holds.

Criticisms and empirical evidence

A number of criticisms have been made with regards to the use and application of Hotelling's lemma in empirical work.

C.Robert Taylor points out that the accuracy of Hotelling's lemma is dependent on the firm maximizing profits, meaning that it is producing profit maximizing output ${\displaystyle y^{*}}$ and cost minimizing input ${\displaystyle x^{*}}$. If a firm is not producing at these optima, then Hotelling's lemma would not hold.[1]

See also

• Hotelling's law
• Hotelling's rule
• Supply and demand
• Shephard's lemma

References

1. Taylor, C. Robert (1989). "Duality, Optimization, and Microeconomic Theory: Pitfalls for the Applied Researcher". Western Journal of Agricultural Economics. 14 (2): 200–212. JSTOR 40988099.

• Hotelling, H. (1932). "Edgeworth's taxation paradox and the nature of demand and supply functions". Journal of Political Economy. 40 (5): 577–616. doi:10.1086/254387. JSTOR 1822600.
• Sakai, Y. (1974). "Substitution and Expansion Effects in Production Theory: The Case of Joint Production". Journal of Economic Theory. 9 (3): 255–274. doi:10.1016/0022-0531(74)90051-9.
• Takayama, A. (1985). Mathematical Economics. New York: Cambridge University Press. pp. 141–144. ISBN 978-0-521-31498-5.
• Varian, H. (1992). Microeconomic Analysis (3rd ed.). New York: W. W Norton. pp. 43–45. ISBN 978-0-393-95735-8.